Partition (Density, Stochastic)

Description

This article describes the Stochastic equation for the partition of particles by size and density in gravity separators.

The Stochastic formulation is applicable to both dense medium separators and gravity concentration processes.

Model theory

Figure 1. Stochastic model partition to sinks. Each coloured line represents a particle class with the geometric mean size indicated in the series label. The pivot point is evident at a density of 2.53 t/m³.

Figure 2. Stochastic model partition to sinks, using the same model parameters as Figure 1. Each coloured line represents a density class with the specific gravity indicated in the series label.

Figure 2. Modified Stochastic model partition to sinks, using the same model parameters as Figure 1 except time, $t$, which is half the original value. Each coloured line represents a density class with the specific gravity indicated in the series label.

Rao et al (2003) proposed a model that describes the size-density partition surface in both dense-medium and gravity separators. The Stochastic model is derived from a statistical consideration of particle settling phenomena, and for a steady-state process is:

$Y_{ij}={\dfrac {1}{2}}\left[1+{\rm {erf}}\left(B-A{{\bar {d}}_{i}}^{c}((\rho _{\rm {S}})_{ij}-\rho _{\rm {p}})\right)\right]$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,q\}$, $q$ is the number of size intervals
• $j$ is the index of the density class, $j=\{1,2,\dots ,m\}$, $m$ is the number of density classes
• $Y_{ij}$ is the fraction of the mass of particles in size class $i$ and density class $j$ which are partitioned (recovered) to the underflow/concentrate/sinks stream (frac)
• $A$ is a model parameter that accounts for gravitational and viscous forces of the separation (-)
• $c$ is a model parameter that accounts for turbulence in the separation (-)
• $B$ is a model parameter that accounts for fluid drift force of the separation (-)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
• $\rho _{\rm {S}}$ is the particle density (kg/m³)
• $\rho _{\rm {p}}$ is a model parameter representing the pivot density of separation (kg/m³)

Rao (2021) extended the Stochastic model to represent the partition surface of an entire batch jig bed in terms of both slice (cut) height and jigging time. The modified Stochastic model is:

${\begin{array}{c}Y_{ij}={\dfrac {1}{2}}\left[1+{\rm {erf}}\left({\dfrac {(h_{\rm {l}}-h_{\rm {f}})}{tD}}+B-A{{\bar {d}}_{i}}^{c}((\rho _{\rm {S}})_{ij}-\rho _{\rm {p}})\right)\right]&{\text{for }}0\leq h_{\rm {l}}\leq 1{\text{ and }}0\leq t\leq t_{\rm {eq}}\\\end{array}}$

where:

• $h_{\rm {l}}$ is the relative slice position of the bed, where 0 is the bottom of the bed and 1 is the top of the bed (m/m)
• $h_{\rm {f}}$ is the feed particle input bed position, a model parameter (m/m)
• $D$ is a model parameter that accounts for the standard deviation of particle velocities in the separator (-)
• $t$ is thje jigging time of the particles (s)
• $t_{\rm {eq}}$ is the jigging time required to attain dynamic equilibrium (s)

The modified Stochastic model reduces to the original formulation when $(h_{\rm {l}}-h_{\rm {f}})\to 0$ or $t\to \infty$, i.e. steady-state.

Rao (2021) does not indicate whether the modified Stochastic model is applicable to dense-medium or gravity concentration processes other than batch jigging. The modified Stochastic model should be applied with caution in such cases.

Partition Metrics

Several metrics are provided to characterise the Stochastic model partition curve.

The $d_{50}$, also known as the cut or separation size, is defined as the size of a particle of a given density which has an even (50%) chance of appearing in either the sinks (concentrate) or floats (tail) stream. The $d_{50}$ size is obtained by rearranging the modified Stochastic model:

$d_{50}=\left({\dfrac {{\frac {(h_{\rm {l}}-h_{\rm {f}})}{tD}}+B}{A(\rho -\rho _{\rm {p}})}}\right)^{\frac {1}{c}}$

The $\rho _{50}$, also known as the cut or separation density, is defined as the density of a particle of a given size which has an even (50%) chance of appearing in either the sinks (concentrate) or floats (tail) stream. The $\rho _{50}$ density is obtained by rearranging the modified Stochastic model:

$\rho _{50}=\rho _{\rm {p}}+\left({\dfrac {{\frac {(h_{\rm {l}}-h_{\rm {f}})}{tD}}+B}{Ad^{c}}}\right)$

The Ecart Probable, or $E_{\rm {p}}$, is a measure of the deviation of a partition curve from a perfect separation, and is found as:

$E_{\rm {p}}=-{\dfrac {0.476936}{Ad^{c}}}$

Finally, the partition coefficient at the pivot point, $Y_{\rm {p}}$ (frac), is:

$Y_{\rm {p}}={\dfrac {1}{2}}\left[1+{\rm {erf}}\left({\dfrac {(h_{\rm {l}}-h_{\rm {f}})}{tD}}+B)\right)\right]$

As with the modified Stochastic model, the partition metrics above reduce to their steady-state counterparts as $(h_{\rm {l}}-h_{\rm {f}})\to 0$ or $t\to \infty$.

Middlings

Gravity concentrators such as jigs, spirals and shaking tables produce a bed or band of partially stratified components at the point of discharge. Portions of the bed or band are then typically directed to product streams by a physical device, such as a weir, 'splitter' or 'cutter'. These devices are usually adjustable, and can be arranged to recover arbitrary fractions of the bed or band.

From a physical standpoint, adjusting the discharge device to recover more of the bed or band has the effect of recovering the both the portion from the previous position plus the portion in between the previous and new positions. As more mass is recovered by this process, the partition curve effectively 'shifts upwards'. The partition curve is thus representing the cumulative recovery of mass from all positions between the beginning of the bed/band and the discharge device position.

Mathematically, the partition curve generated by such a gravity concentration method should also be considered a cumulative recovery of mass to concentrate. When multiple product streams exist, e.g. concentrate and middlings, the partition of components to each individual product stream will be the difference between the cumulative partition curves at each product stream position. That is,

$y_{ij,s}={\begin{cases}Y_{ij,s}&s=1\\\max \left(0,Y_{ij,s}-Y_{ij,s-1}\right)&s>1\\\end{cases}}$

where:

• $s$ is the index of the product stream, i.e. $s=1$ is the first concentrate stream, $s>1$ are subsequent lower-grade concentrate or middlings streams
• $y_{ij,s}$ is the mass fraction of particles in the feed stream in size class $i$ and density class $j$ which are partitioned to the product stream $s$ (frac)
• $Y_{ij,s}$ is the cumulative mass fraction of particles in the feed stream in size class $i$ and density class $j$ which are partitioned to all the products streams up to and including $s$ (frac)
• The $\max$ term prevents negative partition values

The cumulative recovery formulation described above is physically and mathematically distinct from staged recovery processes which apply partition curves to the unrecovered streams of previous partition steps.

Excel

The Stochastic partition equation may be invoked from the Excel formula bar with the following function call:

=mdPartition_Stochastic(MeanSize as Range, RD as Range, A as Double, c as Double, B as Double, rhop as Double, Optional hf as Double = 0, Optional D as double = 0, Optional hl as Double = 0, Optional t as Double = 0, Optional returnPartitionMetrics as Bool = false)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and model results are defined below in matrix notation, along with an example image showing the selection of the same cells in the Excel interface:

${\begin{aligned}{\mathit {MeanSize}}&={\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\\\end{bmatrix}}\\\\{\mathit {RD}}&={\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}\\\\A&={\big [}A{\big ]}\\B&={\big [}B{\big ]}\\c&={\big [}c{\big ]}\\{\mathit {rhop}}&={\big [}\rho _{\rm {p}}{\text{ (t/m}}^{3}{\text{))}}{\big ]}\\{\mathit {hf}}&={\big [}h_{\rm {f}}{\text{ (m/m)}}{\big ]}\\{\mathit {hl}}&={\big [}h_{\rm {l}}{\text{ (m/m)}}{\big ]}\\D&={\big [}D{\big ]}\\t&={\big [}t{\text{ (s)}}{\big ]}\\{\mathit {returnPartitionMetrics}}&={\big [}({\text{True / False)}}{\big ]}\\\\\\{\mathit {mdPartition\_StochasticRao}}&={\begin{bmatrix}{\begin{bmatrix}Y_{11}{\text{ (t/h)}}&\dots &Y_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\Y_{n1}{\text{ (t/h)}}&\dots &Y_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(\rho _{50})_{1}{\text{ (t/m}}^{3}{\text{)}}\\\vdots \\(\rho _{50})_{n}{\text{ (t/m}}^{3}{\text{)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {p}})_{1}{\text{ (t/m}}^{3}{\text{)}}\\\vdots \\(E_{\rm {p}})_{n}{\text{ (t/m}}^{3}{\text{)}}\\\end{bmatrix}}&{\begin{array}{c}{\begin{bmatrix}(d_{50})_{1}{\text{ (mm)}}&\dots &(d_{50})_{m}{\text{ (mm)}}\\\end{bmatrix}}^{\rm {*}}\\\\\\\\\end{array}}\end{bmatrix}}\end{aligned}}$ where: $n$ is the number of size intervals $m$ is the number of density classes $^{*}$ indicates optional results returned if ${\mathit {returnPartitionMetrics=True}}$

Figure 2. Example showing the selection of the input parameters (purple, green, pink, brown, teal, blue, red and purple cells), and the MeanSize (blue frame), RD (red frame) and Results (light blue frame) arrays in Excel. The parameter ${\mathit {returnPartitionMetrics}}={\text{True}}$ in this example.

SysCAD

The Stochastic partition is available from the MetDynamics*GravityConcentrator and MetDynamics*DenseMedium unit models.

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

Note that a Con and Partition page is provided provided for each connected concentrate discharge stream.

MD_GravityConcentrator page

The first tab page in the access window will have this name.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Tag	Display	This name tag may be modified with the change tag option.
Condition	Display	OK if no errors/warnings, otherwise lists errors/warnings.
ConditionCount	Display	The current number of errors/warnings. If condition is OK, returns 0.
GeneralDescription / GenDesc	Display	This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here. If this field is blank, then SysCAD will display the unit class ID.
Requirements
On	CheckBox	This enables the unit. If this box is not checked, then the MassFracToCon option appears below.
MassFracToCon	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the concentrate stream when the model is off.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
SizeForPassingFracCalc	Input	Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc	Input	Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section.
Stream Summary
MassFlow / Qm	Display	The total mass flow in each stream.
SolidMassFlow / SQm	Display	The Solids mass flow in each stream.
LiquidMassFlow / LQm	Display	The Liquid mass flow in each stream.
VolFlow / Qv	Display	The total Volume flow in each stream.
Temperature / T	Display	The Temperature of each stream.
Density / Rho	Display	The Density of each stream.
SolidFrac / Sf	Display	The Solid Fraction in each stream.
LiquidFrac / Lf	Display	The Liquid Fraction in each stream.
Passing	Display	The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream.
Passes	Display	The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction.

Con page

The Con page is used to specify the required model method and associated input parameters.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Stage
On	Checkbox	This enables the stage. If off, the feed to this stage passes directly to the next stage (or tail) without partition.
Method	Partition (User)	The partition to concentrate for each size interval is defined by the user.
	Partition (Pivot)	The partition to concentrate for each size interval is defined by the Pivot model.
	Partition (Stochastic)	The partition to concentrate for each size interval is defined by the Stochastic model.
	Partition (Bazin)	The partition to concentrate for each size interval is defined by the Bazin model.
	Jig (King)	The partition to concentrate for each size interval is defined by the King jig stratification model.
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
Stochastic
A	Input	Partition surface parameter, gravitation and viscous forces.
B	Input	Partition surface parameter, fluid drift force.
c	Input	Partition surface parameter, turbulence.
PivotDensity / Rhop	Input	Density at pivot point.
hf	Input	Feed particle input bed position.
hl	Input	Slice position of the particle bed.
D	Input	Partition surface parameter, standard deviation of particle velocities.
t	Input	Separation period of particles.
Liquids
LiquidsSeparMethod	Split To Con (User)	Liquids are split to concentrate by a user-defined fraction of liquids in the feed.
	Con Solids Fraction	Sufficient liquids mass is recovered to the concentrate stream to yield the user-defined concentrate solids mass fraction value (if possible).
	Con Liquids Fraction	Sufficient liquids mass is recovered to the concentrate stream to yield the user-defined concentrate liquids mass fraction value (if possible).
ConSolidsFracReqd / Con.SfReqd	Input	Required value of the mass fraction of solids in the concentrate stream. Only visible if Con Solids Fraction is selected.
ConLiquidsFracReqd / Con.LfReqd	Input	Required value of the mass fraction of liquids in the concentrate stream. Only visible if Con Liquids Fraction is selected.
LiqSplitToCon / Con.LiqSplit	Input/Display	The fraction of feed liquids recovered to the concentrate stream.

Partition page

The Partition page is used to specify or display the partition by species and size values.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Distribution
Name	Display	Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount	Display	Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount	Display	Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
CumulativePartition
Method	Model/User	Select model-calculated or user-defined cumulative partition to separate each solids species type.
Density	Display	Density of each solid species.
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
All (All column)	Display	Actual overall cumulative partition to sinkss of all solid species, for each size interval. Excludes solid species not present in the gravity concentrator feed.
CumulativePartition	Display	Cumulative partition to sinkss for each size interval, in each solid species, as determined by the selected model or user defined value. Note: These values are displayed regardless of whether the solid species is present in the gravity concentrator feed or not.
All (All row, All column)	Display	Displays the actual, total, cumulative partition of all solids with a particle size distribution property in the feed to sinkss. Excludes solid species not present in the gravity concentrator feed.
All (All row, per species)	Display	Actual overall cumulative partition to sinkss for each solid species, for all size intervals in that species. Excludes solid species not present in the gravity concentrator feed.
Partition
Method	Model/User	Select model-calculated or user-defined partition to separate each solids species type.
Density	Display	Density of each solid species.
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
All (All column)	Display	Actual overall partition to sinks of all solid species, for each size interval. Excludes solid species not present in the gravity concentrator feed.
Partition	Display	Partition to sinks for each size interval, in each solid species, as determined by the selected model or user defined value. Note: These values are displayed regardless of whether the solid species is present in the gravity concentrator feed or not.
All (All row, All column)	Display	Displays the actual, total, partition of all solids with a particle size distribution property in the feed to sinks. Excludes solid species not present in the gravity concentrator feed.
All (All row, per species)	Display	Actual overall partition to sinks for each solid species, for all size intervals in that species. Excludes solid species not present in the gravity concentrator feed.
PartitionMetrics
d50	Display	Cut size (d50) of separation of each ore species.
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
CutDensity / Rho50	Display	Cut density (Rho50) of all particles in each size interval.
EcartProbable / Ep	Display	Ecart Probable of all particles in each size interval.
PivotPartition / Yp	Display	Partition at pivot point.

MD_DenseMedium page

The first tab page in the access window will have this name.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Tag	Display	This name tag may be modified with the change tag option.
Condition	Display	OK if no errors/warnings, otherwise lists errors/warnings.
ConditionCount	Display	The current number of errors/warnings. If condition is OK, returns 0.
GeneralDescription / GenDesc	Display	This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here. If this field is blank, then SysCAD will display the unit class ID.
Requirements
On	CheckBox	This enables the unit. If this box is not checked, then the MassFracToSinks option appears below.
MassFracToSinks	Input	Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the sinks stream when the model is off.
Method	Partition (User)	The partition to sinks for each size interval is defined by the user.
	Partition (Pivot)	The partition to sinks for each size interval is defined by the Pivot model.
	Partition (Stochastic)	The partition to sinks for each size interval is defined by the Stochastic model.
	Drum (Baguley)	The Baguley dense medium drum model is used to determine the partition of solids to sinks and floats for each size interval.
Options
ShowQFeed	CheckBox	QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQSinks	CheckBox	QSinks and associated tab pages (eg Sp) will become visible, showing the properties of the sinks stream.
ShowQFloats	CheckBox	QFloats and associated tab pages (eg Sp) will become visible, showing the properties of the floats stream.
SizeForPassingFracCalc	Input	Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc	Input	Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section.
Stream Summary
MassFlow / Qm	Display	The total mass flow in each stream.
SolidMassFlow / SQm	Display	The Solids mass flow in each stream.
LiquidMassFlow / LQm	Display	The Liquid mass flow in each stream.
VolFlow / Qv	Display	The total Volume flow in each stream.
Temperature / T	Display	The Temperature of each stream.
Density / Rho	Display	The Density of each stream.
SolidFrac / Sf	Display	The Solid Fraction in each stream.
LiquidFrac / Lf	Display	The Liquid Fraction in each stream.
Passing	Display	The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream.
Passes	Display	The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction.

Stochastic page

The Stochastic page is used to specify the required model method and associated input parameters.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Stochastic
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
A	Input	Partition surface parameter, gravitation and viscous forces.
B	Input	Partition surface parameter, fluid drift force.
c	Input	Partition surface parameter, turbulence.
PivotDensity / Rhop	Input	Density at pivot point.
hf	Input	Feed particle input bed position.
hl	Input	Slice position of the particle bed.
D	Input	Partition surface parameter, standard deviation of particle velocities.
t	Input	Separation period of particles.
Liquids
LiquidsSeparMethod	Split To Sinks (User)	Liquids are split to sinks by a user-defined fraction of liquids in the feed.
	Sinks Solids Fraction	Sufficient liquids mass is recovered to the sinks stream to yield the user-defined sinks solids mass fraction value (if possible).
	Sinks Liquids Fraction	Sufficient liquids mass is recovered to the sinks stream to yield the user-defined sinks liquids mass fraction value (if possible).
SinksSolidsFracReqd / Sinks.SfReqd	Input	Required value of the mass fraction of solids in the sinks stream. Only visible if Sinks Solids Fraction is selected.
SinksLiquidsFracReqd / Sinks.LfReqd	Input	Required value of the mass fraction of liquids in the sinks stream. Only visible if Sinks Liquids Fraction is selected.
LiqSplitToSinks / Sinks.LiqSplit	Input/Display	The fraction of feed liquids recovered to the sinks stream.

Partition page

The Partition page is used to specify or display the partition by species and size values.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
Distribution
Name	Display	Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount	Display	Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount	Display	Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
Partition
Method	Model/User	Select model-calculated or user-defined partition to separate each solids species type.
Density	Display	Density of each solid species.
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
All (All column)	Display	Actual overall partition to sinks of all solid species, for each size interval. Excludes solid species not present in the dense medium separator feed.
Partition	Display	Partition to sinks for each size interval, in each solid species, as determined by the selected model or user defined value. Note: These values are displayed regardless of whether the solid species is present in the dense medium separator feed or not.
All (All row, All column)	Display	Displays the actual, total, partition of all solids with a particle size distribution property in the feed to sinks. Excludes solid species not present in the dense medium separator feed.
All (All row, per species)	Display	Actual overall partition to sinks for each solid species, for all size intervals in that species. Excludes solid species not present in the dense medium separator feed.
PartitionMetrics
d50	Display	Cut size (d50) of separation of each ore species.
Size	Display	Size of each interval in mesh series.
MeanSize	Display	Geometric mean size of each interval in mesh series.
CutDensity / Rho50	Display	Cut density (Rho50) of all particles in each size interval.
EcartProbable / Ep	Display	Ecart Probable of all particles in each size interval.
PivotPartition / Yp	Display	Partition at pivot point.

About page

This page is provides product and licensing information about the Met Dynamics Models SysCAD Add-On.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
About
HelpLink		Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information		Copies Product and License information to the Windows clipboard.
Product
Name	Display	Met Dynamics software product name
Version	Display	Met Dynamics software product version number.
BuildDate	Display	Build date and time of the Met Dynamics Models SysCAD Add-On.
License
File		This is used to locate a Met Dynamics software license file.
Location	Display	Type of Met Dynamics software license or file name and path of license file.
SiteCode	Display	Unique machine identifier for license authorisation.
ReqdAuth	Display	Authorisation level required, MD-SysCAD Full or MD-SysCAD Runtime.
Status	Display	License status, LICENSE_OK indicates a valid license, other messages report licensing errors.
IssuedTo	Display	Only visible if Met Dynamics license file is used. Name of organisation/seat the license is authorised to.
ExpiryDate	Display	Only visible if Met Dynamics license file is used. License expiry date.
DaysLeft	Display	Only visible if Met Dynamics license file is used. Days left before the license expires.

Additional notes

• Solid species that do not possess a particle size distribution property are split according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration.
• If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
• Gas phase species report directly to the tail/floats stream without split.

See also

• Partition (Density, Pivot)